Physical systems can be modeled mathematically to simulate their behavior under certain conditions. There are a wide variety of means to model these systems, ranging from the very simplistic to the extremely complicated. One of the more complicated means to model physical systems is through the use of finite element analysis. As the name implies, finite element analysis involves the representation of individual, finite elements of the physical system in a mathematical model and the solution of this model in the presence of a predetermined set of boundary conditions.
In finite element modeling, the region that is to be analyzed is broken up into sub-regions called elements. This process of dividing the region into sub-regions may be referred to as discretization or mesh generation. The region is represented by functions defined over each element. This generates a number of local functions that are much simpler than those which would be required to represent the entire region. The next step is to analyze the response for each element. This is accomplished by building a matrix that defines the properties of the various elements within the region and a vector that defines the forces acting on each element in the structure. Once all the element matrices and vectors have been created, they are combined into a structure matrix equation. This equation relates nodal responses for the entire structure to nodal forces. After applying boundary conditions, the structure matrix equation can be solved to obtain unknown nodal responses. Intra-element responses can be interpolated from nodal values using the functions which were defined over each element.
Finite element models are often used to determine the behavior of geological structures such as oil reservoirs under certain conditions. Finite element models can simulate the flow of oil through particular regions of the reservoir in response to the various oil recovery operations, such as drilling. The resulting information is useful in the analysis of the reservoir and the management of the oil recovery operations.
Conventional finite element models, however, have certain limitations which prevent them from accurately simulating the behavior of the physical systems which the model. For example, in the case of the oil reservoir, the finite element model may represent a particular property of the reservoir using a mathematical function which gradually changes according to the position within the reservoir. It may be very difficult to force these functions to approximate changes in the properties which occur very abruptly as a result of geological features within the reservoir. For instance, a reservoir may have two distinct layers, each of which has a substantially different value for a given property (e.g., porosity.) at the boundary between these two layers, a mathematical function representing the value of the property might therefore have to make an immediate transition from one value to the other at the boundary between the layers. Even if the mathematical function can be adapted to represent the respective values for the property at the boundary between the two layers, such a function would typically be very complex and would require a great deal of computational resources.
It would therefore be desirable to provide a method for more accurately representing the values of certain properties within a finite element model, particularly across the boundaries of features within the model for which the properties change rapidly or are discontinuous, while at the same time requiring a relatively small amount of computational resources.